math.sqrt on complex, imaginary part, im greater than 0 branch

Average Error: 38.4 → 18.5

Time: 4.2s

Precision: binary64

\[im \gt 0.0\]

Math

\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \le 0.0:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{re + \sqrt{re \cdot re + im \cdot im}}}\\ \mathbf{elif}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \le 7.291187668146529 \cdot 10^{74}:\\ \;\;\;\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{im - re}\right)\\ \end{array}\]

C

double code(double re, double im) {
	return ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) - re))))))));
}

↓

double code(double re, double im) {
	double VAR;
	if ((((double) sqrt(((double) (2.0 * ((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) - re)))))) <= 0.0)) {
		VAR = ((double) (0.5 * ((double) (((double) sqrt(((double) (2.0 * ((double) (im * im)))))) / ((double) sqrt(((double) (re + ((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))))))))))));
	} else {
		double VAR_1;
		if ((((double) sqrt(((double) (2.0 * ((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) - re)))))) <= 7.291187668146529e+74)) {
			VAR_1 = ((double) (((double) sqrt(((double) (2.0 * ((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) - re)))))) * 0.5));
		} else {
			VAR_1 = ((double) (0.5 * ((double) (((double) sqrt(2.0)) * ((double) sqrt(((double) (im - re))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 3 regimes

2. if (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))) < 0.0

   1. Initial program 57.4

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
   2. Using strategy rm

   3. Applied flip-- 57.4

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} + re}}}\]

   4. Applied associate-*r/ 57.4

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}{\sqrt{re \cdot re + im \cdot im} + re}}}\]

   5. Applied sqrt-div 57.4

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}}\]

   6. Simplified 24.8

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{\left(im \cdot im\right) \cdot 2}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\]

   7. Simplified 24.8

      \[\leadsto 0.5 \cdot \frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\color{blue}{\sqrt{re + \sqrt{re \cdot re + im \cdot im}}}}\]

   if 0.0 < (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))) < 7.291187668146529e74

   1. Initial program 4.1

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]

   if 7.291187668146529e74 < (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))

   1. Initial program 62.6

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]

   2. Taylor expanded around 0 29.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{im} - re\right)}\]
   3. Using strategy rm

   4. Applied sqrt-prod 29.1

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{im - re}\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 18.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \le 0.0:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{re + \sqrt{re \cdot re + im \cdot im}}}\\ \mathbf{elif}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \le 7.291187668146529 \cdot 10^{74}:\\ \;\;\;\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{im - re}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  :pre (> im 0.0)
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))